Salt decomposes in water according to the law of uninhibited decay. If the initial amount of salt is 40 kg , and after 8 hours 20 kg of salt is left, find the decay rate k.
The correct answer and explanation is:
To find the decay rate kk for the decomposition of salt in water, we can use the formula for uninhibited (exponential) decay: A(t)=A0e−ktA(t) = A_0 e^{-kt}
Where:
- A(t)A(t) is the amount of substance at time tt,
- A0A_0 is the initial amount of the substance,
- kk is the decay constant (rate),
- tt is the time elapsed.
Given:
- Initial amount: A0=40A_0 = 40 kg
- After t=8t = 8 hours, A(8)=20A(8) = 20 kg
Substitute into the equation: 20=40e−8k20 = 40 e^{-8k}
Divide both sides by 40: 12=e−8k\frac{1}{2} = e^{-8k}
Take the natural logarithm of both sides: ln(12)=ln(e−8k)\ln\left(\frac{1}{2}\right) = \ln\left(e^{-8k}\right) ln(12)=−8k\ln\left(\frac{1}{2}\right) = -8k −ln(2)=−8k- \ln(2) = -8k k=ln(2)8k = \frac{\ln(2)}{8} k≈0.69318≈0.0866 per hourk \approx \frac{0.6931}{8} \approx 0.0866 \text{ per hour}
✅ Final Answer:
k≈0.0866 per hour\boxed{k \approx 0.0866 \text{ per hour}}
💡 Explanation (300 words):
The decay of salt in water, when not influenced by external factors, is modeled by the law of uninhibited decay, which assumes that the rate at which the salt disappears is proportional to how much salt is currently present. This is mathematically described using the exponential decay function: A(t)=A0e−ktA(t) = A_0 e^{-kt}
In our case, the initial salt mass is 40 kg. After 8 hours, only 20 kg remains. This means that half the salt has decayed over that time. By substituting into the decay formula, we solved for the decay constant kk, which determines how quickly the substance is diminishing.
We used logarithmic manipulation to isolate kk, recognizing that ln(ex)=x\ln(e^x) = x, a fundamental property of logarithms. The resulting equation involved the natural logarithm of 2 because half the substance remained. This situation, known as a half-life problem, is common in exponential decay scenarios such as radioactive decay, cooling, or chemical decomposition like salt dissolving.
The final result, k≈0.0866k \approx 0.0866, tells us that about 8.66% of the salt is decaying every hour. This decay rate can now be used to predict how much salt will be left at any future time or how long it will take for the salt to decay to a certain level.